The ratio of the area of a regular hexagon and an equilateral triangle having same perimeter is

Options

A. (a) `2:3`
B. (b) `6:1`
C. (c) `3:2`
D. (d) `1:6`

Answer

To find the ratio of the area of a regular hexagon and an equilateral triangle having the same perimeter, we can follow these steps:

### Step 1: Define the Perimeter
Let the side length of the regular hexagon be \( x \). Since a regular hexagon has 6 equal sides, its perimeter \( P \) can be expressed as:
\[ P = 6x \]

### Step 2: Determine the Side Length of the Equilateral Triangle
For the equilateral triangle to have the same perimeter, let the side length of the equilateral triangle be \( y \). The perimeter of the equilateral triangle is:
\[ P = 3y \]
Since both perimeters are equal, we can set them equal to each other:
\[ 6x = 3y \]
From this equation, we can solve for \( y \):
\[ y = 2x \]

### Step 3: Calculate the Area of the Regular Hexagon
The area \( A_h \) of a regular hexagon can be calculated using the formula:
\[ A_h = \frac{3\sqrt{3}}{2} s^2 \]
where \( s \) is the side length of the hexagon. Substituting \( s = x \):
\[ A_h = \frac{3\sqrt{3}}{2} x^2 \]

### Step 4: Calculate the Area of the Equilateral Triangle
The area \( A_t \) of an equilateral triangle can be calculated using the formula:
\[ A_t = \frac{\sqrt{3}}{4} s^2 \]
where \( s \) is the side length of the triangle. Substituting \( s = 2x \):
\[ A_t = \frac{\sqrt{3}}{4} (2x)^2 = \frac{\sqrt{3}}{4} \cdot 4x^2 = \sqrt{3} x^2 \]

### Step 5: Find the Ratio of the Areas
Now, we can find the ratio of the area of the hexagon to the area of the triangle:
\[ \text{Ratio} = \frac{A_h}{A_t} = \frac{\frac{3\sqrt{3}}{2} x^2}{\sqrt{3} x^2} \]
The \( x^2 \) terms cancel out:
\[ \text{Ratio} = \frac{\frac{3\sqrt{3}}{2}}{\sqrt{3}} = \frac{3}{2} \]

### Step 6: Conclusion
Thus, the ratio of the area of the regular hexagon to the area of the equilateral triangle is:
\[ \text{Ratio} = \frac{3}{2} \]